The non-destructive investigation of samples is an important object in various technical fields like material sciences, medical examinations, archaeology, construction technique, techniques concerning security matters etc. One approach for obtaining an image of a sample e.g. by computer tomography (CT) is based on an irradiation trough a object plane from different projection directions with X-rays, followed by a reconstruction of the object plane on the basis of attenuation data measured at different directions. The irradiation of a region of investigation 2′ with a conventional fan beam 5′ created by an X-ray source 210′ is schematically illustrated in FIG. 9. The fan beam 5′ comprises a continuous distribution of electro-magnetic fields shaped according to an emission characteristic of the X-ray source 210′. The entirety of the attenuation data measured with a detector 310′ can be described in terms of so-called Radon data in a Radon space.
The most relevant conventional reconstruction methods known today can be summarized as methods based on the iterative reconstruction or those based on the so-called filtered back-projection. The iterative reconstruction methods have essential disadvantages in terms of extremely long calculation times. On the other hand, the filtered back-projection method has a general disadvantage as an interpolation step included in the reconstruction results in errors and artifacts which have a tendency even to increase with increasing space frequency. Another problem of the filtered back-projection method is related to the discretization of the Radon data from which the image data have to be reconstructed. To get an optimal filtered back-projection reconstruction it would be necessary to exactly match the projected irradiation rays with detector elements of a detector. This is in general not the case. For this reason, uncertainties or smoothing effects from the reconstruction of Radon data by means of filtered back-projection algorithms are introduced.
T. Bortfeld et al. have described a so-called Chebyshev domain filtered back projection (CD-FBP) algorithm for the reconstruction of two-dimensional images from a plurality of projections along the projection directions (“Phys. Med. Biol.”, Vol. 44, 1999, p. 1105-1120). With this CD-FBP algorithm, the projections are represented as decompositions, which are subjected to the above filtered back-projection reconstruction. The projections are measured e.g. with a fan beam geometry, wherein attenuation values according to single projection lines with even angular intervals relative to each other are measured. The single projection lines measured with different projection directions of the fan beam can be resorted for providing parallel projections to be used for the image reconstruction. The CD-FBP algorithm has not yielded a practical implementation. The algorithm assumes an ideal fan beam geometry, which is not available in practice. Therefore, the T. Bortfeld et al. algorithm requires an interpolation step like the conventional filtered back-projection. Furthermore, as the CD-FBP algorithm is inherently discrete, there is a lack of adaptation to the continuous radiation characteristic of conventional radiation sources. Finally, the CD-FBP algorithm has an essential disadvantage in terms of artifacts occurring in the reconstructed images.
The above disadvantages are associated not only with the conventional CT imaging, but also with all available reconstruction methods related to Radon data.